Kr,s Graph Bootstrap Percolation

Abstract

A graph G percolates in the Kr,s-bootstrap process if we can add all missing edges of G in some order such that each edge creates a new copy of Kr,s, where Kr,s is the complete bipartite graph. We study Kr,s-bootstrap percolation on the Erdős-Rényi random graph, and determine the percolation threshold for balanced Kr,s up to a logarithmic factor. This partially answers a question raised by Balogh, Bollobás, and Morris. We also establish a general lower bound of the percolation threshold for all Kr,s, with r ≥ s ≥ 3.